[↗] GIDS series

preprint · appendix a · 6 of 7

Appendix A: Study Guide / Cheat Sheet

This appendix is the shorter working guide to the manuscript. The complete source of truth is Canonical Notation and Mathematical Conventions. When the shorthand here and the canonical file appear to disagree, use the canonical file.

The whole program in one view

The philosophical arc is

\mathcal N \xrightarrow{\;\operatorname{Reg}\;} \mathcal G \longrightarrow G_i \longrightarrow T_{i,t} \longrightarrow \phi_{i,t}.

Read this as:

there is an external domain larger than any actor’s experience;
a model registers some external configuration in GIDS;
an actor inherits a restricted organization of possible distinctions;
development and history realize that organization in one actor;
the actor occupies a full phenomenal state at a time.

For complete one-step statements, use

\Sigma_{i,t}^{\star} = (T_{i,t},\phi_{i,t},c_{i,t},w_t), \qquad \Sigma_{i,t+1}^{\star} \sim K_i^{\star} \!\left( \cdot \mid \Sigma_{i,t}^{\star},X_t=x_t, \boldsymbol\Xi_{t+1}=\boldsymbol\xi_{t+1} \right).

The predictive arc is

\mathsf I_{i,t} \longrightarrow Q_{i,t} \xrightarrow{\;\Pi_{\tau,\Delta}\;} q_{i,t}^{(\tau,\Delta)},

where \(Q_{i,t}\) is the actor’s general predictive response object and \(q_{i,t}^{(\tau,\Delta)}\) is a task-and-horizon summary when such a sufficient summary exists.

The operational arc for one person is

\mathcal H_{i,<t} \longrightarrow \widehat s_{i,t} = \left( \widehat{\mathbf t}_{i,t}, \mathbf z_{i,t}, \mathbf c_{i,t}, \mathbf w_t \right).

The first application uses a dyadic state:

\widehat D_{ab,t} = \left( \widehat s_{a,t}^{\mathrm{person}}, \widehat S_{C_a,t}, \widehat s_{b,t}^{\mathrm{person}}, \widehat S_{C_b,t}, \widehat\Gamma_{ab,t}, \mathbf w_t \right).

The learned transition and filtering loop is

\widetilde D_{ab,t+1} \sim K_{D,\theta} \left( \cdot \mid \widehat D_{ab,t}, X_t=x_t, \boldsymbol\Xi_{t+1}=\boldsymbol\xi_{t+1} \right),

\widetilde O_{t+1} \sim R_{O,\theta} \left( \cdot \mid \widetilde D_{ab,t+1} \right),

\widehat D_{ab,t+1} = \mathcal F_\theta \left( \widehat D_{ab,t}, x_t, \mathcal H_{(t,t+1]} \right).

The first equation simulates. The second emits an immediate trace. The third updates the estimate after real records arrive. They are not interchangeable.

1. What GIDS is

Let \(\mathcal N\) be the external or noumenal domain. The manuscript does not assume that \(\mathcal N\) itself is a vector space.

Let

\mathcal G

be God’s Infinite Dimensional Space: an idealized real separable Hilbert space of possible distinctions that could enter the experience or response of an actor.

A registration map

\operatorname{Reg}:\mathcal N_{\mathrm{loc}}\to\mathcal G

turns a local external configuration into a model-side representation:

\mathbf g_t^{\mathrm{reg}}=\operatorname{Reg}(\omega_t).

The representation is not the external object. The map \(\operatorname{Reg}\) may lose information, identify physically different configurations, and depend on the modeling resolution.

Why a Hilbert space is useful

A Hilbert space supplies:

coordinates and inner products;
limits of convergent sequences;
finite-dimensional subspaces inside an open-ended ambient arena;
orthogonal projection when that particular approximation is justified;
room for distinctions unavailable to the actor currently being modeled.

It does not prove that cognition is linear, that psychology has one natural Euclidean chart, or that every meaningful category is a basis vector.

2. Lineage access and the inherited seed

A finite-dimensional lineage idealization is

\mathcal M^{\mathrm{spec}} = \operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_d\} \subset\mathcal G.

If the basis vectors are orthonormal, the corresponding projection is

P^{\mathrm{spec}}\mathbf g = \sum_{j=1}^{d} \langle\mathbf v_j,\mathbf g\rangle\mathbf v_j.

This is the closest point in \(\mathcal M^{\mathrm{spec}}\) under the chosen Hilbert norm. That geometric fact does not establish that an organism literally performs an orthogonal projection.

The inherited seed of individual \(i\) is

G_i = \left( \mathcal M_i^0, A_i^0, \mathcal I_i^0 \right), \qquad A_i^0:\mathcal G\to\mathcal M_i^0.

Here \(A_i^0\) is an individual access map and \(\mathcal I_i^0\) is an inherited starting organization. The access map is not assumed linear or orthogonal.

3. Objects and categories are constructed through the space

A person, proposition, corporation, table, threat, or category is not automatically one primitive vector in GIDS.

For actor \(i\), an external configuration \(\omega_t\) can induce an actor-relative representation

\zeta_{i,t}^{\mathrm{obj}}(\omega_t) = \mathcal I_{i,t} \left( A_{i,t}\operatorname{Reg}(\omega_t), \phi_{i,t}, c_{i,t} \right).

The access codomain \(\mathcal V_{i,t}^{\mathrm{acc}}\) and object codomain \(\mathcal V_{i,t}^{\mathrm{obj}}\) are actor-relative spaces. Neither needs to be a linear subspace of \(\mathcal G\).

A category requires a declared representation domain \(\mathcal V_\kappa^{\mathrm{cat}}\), which may be \(\mathcal G\), an actor-relative object space, or a finite learned feature space. It may then be represented as:

a measurable region \(\mathcal C_\kappa\subseteq\mathcal V_\kappa^{\mathrm{cat}}\);
a prototype point \(\operatorname{proto}_\kappa\in\mathcal V_\kappa^{\mathrm{cat}}\);
a probability measure \(\mathbb P_\kappa\) on \(\mathcal V_\kappa^{\mathrm{cat}}\);
a score \(\operatorname{cat}_{\kappa}:\mathcal V_\kappa^{\mathrm{cat}}\to[0,1]\);
or a learned relational object.

Only a prototype is literally a point of its declared domain. Regions are subsets; distributions and scores are mathematical objects defined on the domain. A category is therefore not required to be a vector or a primitive part of GIDS.

Valid and invalid algebra

Valid operations are operations with a declared interpretation, such as:

projecting a registered signal into an accessible subspace;
calculating similarity after the encoders have been calibrated for that use;
aggregating structured member states into a corporate state;
applying a transition kernel;
integrating expected utility over predicted futures.

Invalid by default:

adding a person vector to an email vector and calling the sum a person;
averaging employees and calling the result a corporation;
treating every dot product as psychological compatibility;
assuming that proximity in a learned chart is observer-independent truth.

A common arena gives us a grammar for constructing relations. It does not bless arbitrary arithmetic.

4. The person hierarchy

The slowly changing realized person is

T_{i,t} = \mathcal E_{\mathrm{ind}} \left( G_i, \mathbf u_{i,<t}^{\mathrm{lang}}, \mathcal H_{i,<t}^{\mathrm{life}} \right).

The full phenomenal state is

\phi_{i,t}\in\Phi_i.

The person-in-role object, the Chimera, is

\chi_{i,t}=(T_{i,t},c_{i,t}).

The hierarchy matters:

\(G_i\) is inherited starting structure;
\(T_{i,t}\) is the realized and slowly changing person;
\(\phi_{i,t}\) is the complete current lived state;
\(Q_{i,t}\) is the general predictive response object;
\(q_{i,t}^{(\tau,\Delta)}\) is a task summary when it exists;
\(\widehat s_{i,t}\) is the finite state estimated by the model.

These are not six names for the same thing.

5. Factor analysis and “fundamental features”

Classical factor analysis writes an observed variable vector as

\mathbf x_i = \boldsymbol\mu + \boldsymbol\Lambda_{\mathrm{FA}}\mathbf f_i + \boldsymbol\varepsilon_i, \qquad \mathbb E[\mathbf f_i]=\mathbf 0, \qquad \operatorname{Cov}(\mathbf f_i)=I, \qquad \mathbb E[\boldsymbol\varepsilon_i\mid\mathbf f_i]=\mathbf 0.

Factor-analytic programs helped produce useful personality constructs by compressing correlations among many observations. Those constructs are often coarse conglomerations over subtler distinctions and response tendencies; their interpretation depends on the measurement design and identification or rotation convention.

GIDS uses the same broad discovery instinct but asks for finer and more operational structure:

persistent factors;
fast state;
role and environment;
proposition-conditioned activation;
transition prediction;
cross-task transfer;
intervention tests.

A learned factor earns the word “fundamental” only provisionally. It should persist where persistence is expected, transfer across contexts, improve multiple predictions, survive ablation, and—when possible—participate in successful intervention. Rotation and reparameterization mean the coordinate label itself is rarely sacred; the predictive information is what matters.

6. General predictive response state

Define the ideal pre-proposition information state

\mathsf I_{i,t} = \left( \mathcal H_{i,<t}, T_{i,t}, c_{i,t}, w_t \right).

This is the reference information bundle for prediction, not the complete actor–world state; direct access to \(\phi_{i,t}\) remains unavailable.

For a finite sequence horizon \(H\), proposition path \(\mathbf x_{t:t+H-1}\), and supplied exogenous scenario \(\boldsymbol\xi_{t+1:t+H}\), choose a version of the observational response kernel

\mathscr R_{i,t}^{(H),\mathrm{obs}} \left( \cdot \mid \mathsf I_{i,t}, \mathbf x_{t:t+H-1}, \boldsymbol\xi_{t+1:t+H} \right) := \mathcal L \left( O_{i,t+1:t+H} \mid \mathsf I_{i,t}, \mathbf X_{t:t+H-1}=\mathbf x_{t:t+H-1}, \boldsymbol\Xi_{t+1:t+H}=\boldsymbol\xi_{t+1:t+H} \right).

This kernel is defined up to almost-sure equality and is operationally used on the support of the proposition and scenario process. For adaptive future propositions, the factual joint law must include the historical policy and candidate-set process. Merely indexing the observational kernel by a new policy does not identify that policy’s counterfactual law; controlled kernels plus an identification argument, or an explicit extrapolation assumption, are required. A causal response object replaces historical conditioning with the corresponding interventional laws.

Two ideal information states are equivalent when they induce the same declared family of future response laws. This defines an abstract predictive information object. A convenient measurable quotient is not automatic: when one exists, denote it by \(Q_{i,t}\); otherwise, use the indexed response-kernel family itself. Either object may be infinite-dimensional.

For task \(\tau\) and elapsed-time horizon \(\Delta\), write

q_{i,t}^{(\tau,\Delta)} = \Pi_{\tau,\Delta}(Q_{i,t})

only when a sufficient task summary exists.

The decision-associated outcome is indexed by the decision, not by pretending the label is an immediate next state:

Y_{i,t}^{(\tau,\Delta)}.

For every measurable event \(B\), sufficiency means

\mathbb P \left( Y_{i,t}^{(\tau,\Delta)}\in B \mid \mathsf I_{i,t},X_t=x \right) = \mathbb P \left( Y_{i,t}^{(\tau,\Delta)}\in B \mid q_{i,t}^{(\tau,\Delta)},X_t=x \right).

This is observational predictive sufficiency unless the laws are explicitly interventional.

7. Learned slow and fast state

The operational person-state is

\widehat s_{i,t} = \left( \widehat{\mathbf t}_{i,t}, \mathbf z_{i,t}, \mathbf c_{i,t}, \mathbf w_t \right).

\(\widehat{\mathbf t}_{i,t}\) is the slow learned person vector;
\(\mathbf z_{i,t}\) is the fast latent state;
\(\mathbf c_{i,t}\) is role and institutional context;
\(\mathbf w_t\) is measured world state.

Let \(v_n=\operatorname{time}(\mathsf h_n)\). If the chronological record stream is shared across actors, let \(\mathbf d_{i,n}^{\mathrm{rec}}\) indicate whether record \(n\) pertains to actor \(i\) and which actor-specific fields are available. The fast state is updated after chronological record \(n\) becomes available:

\mathbf z_{i,n} = U_{z,\theta} \left( \mathbf z_{i,n-1}, \widehat{\mathbf t}_{i}(v_n^-), \mathbf c_i(v_n^-), \mathbf w(v_n^-), \mathsf h_n, \mathbf d_{i,n}^{\mathrm{rec}} \right).

The applicability vector is not a memory representation and not a label mask. It contains a binary applicability flag; when that flag is zero, the fast-state update is the identity.

At decision \(t\),

N(t)=\#\{n:v_n<\operatorname{time}(\mathsf d_t)\}, \qquad \mathbf z_{i,t}=\mathbf z_{i,N(t)}.

The response caused by \(x_t\) therefore cannot already be inside \(\mathbf z_{i,t}\).

Relevance and salience

The ideal task relevance map is

\Lambda_\tau \left( T_{i,t}, \phi_{i,t}, c_{i,t}, w_t, x_t \right).

Salience is

\boldsymbol\lambda_{i,t}^{(\tau)}(x_t) \in[0,1]^{d_\tau},

and the active slice is

\mathbf r_{i,t}^{(\tau)}(x_t) = \boldsymbol\lambda_{i,t}^{(\tau)}(x_t) \odot \Lambda_\tau \left( T_{i,t}, \phi_{i,t}, c_{i,t}, w_t, x_t \right).

The symbol \(\mathbf z_{i,t}\) is not reused for this active slice.

8. What approximation means

The manuscript no longer writes \(s\approx q\) and leaves the symbol unexplained.

If \(\widehat s_{i,t}\) is constructed from the full pre-proposition information state, define the information lost by compression as

\epsilon_{\tau,\Delta}^{\mathrm{state}}(\widehat s) = I \left( Y_{i,t}^{(\tau,\Delta)}; \mathsf I_{i,t} \mid \widehat s_{i,t},X_t \right).

This equals zero when the operational state retains all observational predictive information about the outcome that was available in the ideal information state, conditional on the proposition.

Separately, for a fitted conditional law \(P_{Y,\theta,\tau,\Delta}\), define model-fitting error by

\epsilon_{\theta,\tau,\Delta}^{\mathrm{model}}(\widehat s) = \mathbb E \left[ D_{\mathrm{KL}} \left( \mathbb P(Y\in\cdot\mid\widehat s_{i,t},X_t) \,\middle\|\, P_{Y,\theta,\tau,\Delta}(\cdot\mid\widehat s_{i,t},X_t) \right) \right].

The first error asks whether the state threw information away. The second asks whether the fitted predictor used the retained information correctly. A model can fail either way.

Under log score, these gaps add exactly to excess risk over the full-information Bayes log risk when the conditional laws admit densities or mass functions with respect to one fixed reference measure and the required expectations are finite. For discrete outcomes that Bayes risk is conditional entropy. For continuous outcomes it is expected negative log density, not an invariant entropy of the actor-state.

9. Memory and categorical traces

A tractable memory field is

\mathbf m_{i,t}^{\mathrm{mem}} = \sum_{j=1}^{N_i} \varpi_{ij,t}\mathbf h_{ij}^{\mathrm{mem}}.

This is a model of weighted traces, not a literal claim about storage in the brain.

Categorical observations are indexed by feature family \(f\), source channel \(\sigma\), and role or regime \(\rho\). Contextual lifting occurs before pooling so that surface contradictions can be retyped rather than averaged into nonsense.

Slow categorical memory summarizes durable evidence available before the current decision. Fast categorical memory emphasizes recent and task-relevant evidence. Missingness uses explicit masks and learned null representations rather than pretending that absence equals numerical zero.

10. Corporations as composite actors

A corporation is not a point obtained by averaging its employees.

Let \(\mathcal J_C(t)\) be the people relevant to corporation \(C\)‘s decision at time \(t\), and define

\varsigma_{j,t}^{\mathrm{person}} = \bigl(T_{j,t},\phi_{j,t},c_{j,t}\bigr).

The conceptual composite is

S_{C,t} = \mathcal A_{\mathrm{corp}} \left( \bigl(\varsigma_{j,t}^{\mathrm{person}}\bigr)_{j\in\mathcal J_C(t)}, \mathsf{Facts}_{C,t}, \mathsf{Org}_{C,t}, \mathsf{Mem}_{C,t}, \mathsf{Inc}_{C,t}, \mathsf{EnvHist}_{C,t} \right).

The measurable company-side evidence is

\mathbf u_{C,t}^{\mathrm{corp}} = \left[ \mathbf f_{C,t}^{\mathrm{corp}}, \mathbf g_{C,t}^{\mathrm{org}}, \mathbf m_{C,t}^{\mathrm{inst}}, \boldsymbol\iota_{C,t}^{\mathrm{inc}}, \mathbf h_{C,t}^{\mathrm{env}} \right],

and a learned estimate is

\widehat S_{C,t} = \mathcal A_{\mathrm{corp},\theta} \left( \bigl(\widehat s_{j,t}^{\mathrm{person}}\bigr)_{j\in\mathcal J_C^{\mathrm{obs}}(t)}, \mathbf u_{C,t}^{\mathrm{corp}}, \mathbf m_{C,t}^{\mathrm{miss}} \right).

The terms encode relevant people, facts and statistics, authority and communication structure, institutional memory, incentives and constraints, environmental history, and explicit missingness.

The aggregator must be capable of representing veto rights, coalitions, unequal authority, and information bottlenecks. This supports treating the corporation as an actor for prediction. It does not, by itself, claim human-like consciousness.

11. The sales dyad

The first concrete state is

\widehat D_{ab,t} = \left( \widehat s_{a,t}^{\mathrm{person}}, \widehat S_{C_a,t}, \widehat s_{b,t}^{\mathrm{person}}, \widehat S_{C_b,t}, \widehat\Gamma_{ab,t}, \mathbf w_t \right).

Here:

\(a\) is the salesperson;
\(b\) is the executive;
\(C_a\) and \(C_b\) are their companies;
\(\widehat\Gamma_{ab,t}\) is relationship state;
\(\mathbf w_t\) is shared world state.

A proposition may contain content, offer, price, framing, evidence, channel, timing, sender, and requested action. The dyadic actor-relative proposition and interaction maps are

\mathbf p_{b,t}^{(D)}(x_t) = \mathcal P_{D,\theta} \left( \mathbf e_t^x, \widehat D_{ab,t} \right),

\mathbf h_{ab,t}^{(D,\mathrm{int})}(x_t) = \Psi_{D,\theta} \left( E_{D,\theta}(\widehat D_{ab,t}), \mathbf p_{b,t}^{(D)}(x_t) \right).

These are distinct from the individual-state maps \(\mathcal P_{s,\theta}\) and \(\Psi_{s,\theta}\). The recursively closed transition is

\widetilde D_{ab,t+1} \sim K_{D,\theta}^{\mathrm{int}} \left( \cdot \mid \mathbf h_{ab,t}^{(D,\mathrm{int})}(x_t), \boldsymbol\xi_{t+1} \right).

It is recursively closed because the output is the same state type required at the next step. Evaluating the parameterized kernel at a supplied \(\boldsymbol\xi_{t+1}\) gives a scenario forecast; this does not require a positive-probability singleton event. An unconditional forecast integrates the conditional transition against a declared law for \(\boldsymbol\Xi_{t+1}\).

12. Event time and labels

The pre-decision history is

\mathcal H_{<t} = \left( \mathsf h_n: \operatorname{time}(\mathsf h_n) < \operatorname{time}(\mathsf d_t) \right).

The decision record is

\mathsf d_t = \left( \operatorname{id}_t, \mathcal X_t^{\mathrm{cand}}, x_t, \delta_t, \mathbf e_t^{\mathrm{cat}}, \eta_t, \mathsf I_t^\mu \right),

with logged assignment probability or density

\eta_t = \mu_t \left( x_t \mid \mathsf I_t^\mu, \mathcal X_t^{\mathrm{cand}} \right).

Primary outcomes are

\mathbf Y_t = \left( Y_t^{(\tau_\ell,\Delta_\ell)} \right)_{\ell=1}^{L_Y},

and auxiliary probes are

\mathbf Z_t = \left( Z_t^{(m,\Delta_m)} \right)_{m=1}^{M_Z}.

The label mask is

R_{t,\ell}^{\mathrm{obs}} = \mathbf 1 \{Y_t^{(\tau_\ell,\Delta_\ell)} \text{ is observed by the analysis cutoff}\}.

A missing 90-day label is not a negative. It is censored or unavailable.

Probe labels need their own masks:

R_{t,m}^{Z,\mathrm{obs}} = \mathbf 1\{Z_t^{(m,\Delta_m)}\text{ is observed by the analysis cutoff}\}.

13. Forecasting, ranking, and causality

For a score extending beyond the immediate next response, declare an evaluation regime \(\mathfrak e\) containing the continuation policy, future candidate-set process, exogenous-path law \(\mathbb Q_{\Xi}\), and the outcome/censoring convention. The predictive value is

V_{\theta,\mathfrak e}^{\mathrm{pred}}(x\mid\widehat D_{ab,t}) = \mathbb E_{\theta,\mathfrak e} \left[ U_\tau \left( \widetilde D_{ab,t+1:t+H}, \widetilde O_{t+1:t+H}, \widetilde{\mathbf Y}_t, \widetilde{\mathbf Z}_t \right) \mid \widehat D_{ab,t},X_t=x \right].

The outcome and probe bundles must be functionals of the same rollout or draws from a coherent joint law \(P_{YZ,\theta,\mathfrak e}\). If only separate marginal heads are fitted, the utility may use marginal expectations or an explicitly declared coupling; separate heads do not silently imply independence. When a maximum is attained over a nonempty available set \(\mathcal X_t^{\mathrm{cand}}\subseteq\mathcal X_t^{\mathrm{adm}}\), ranking chooses

x_t^\star \in \arg\max_{x\in\mathcal X_t^{\mathrm{cand}}} V_{\theta,\mathfrak e}^{\mathrm{pred}}(x\mid\widehat D_{ab,t}).

A nonempty finite candidate set guarantees an attained maximum. For an infinite candidate set, use conditions such as compactness and upper semicontinuity, or replace the argmax with a supremum and an approximate optimizer. This is a forecast under the fitted model. The corresponding causal value under the same regime is

V_{\mathfrak e}^{\mathrm{causal}}(x\mid\widehat D_{ab,t}) = \mathbb E_{\mathfrak e} \left[ U_\tau \mid \widehat D_{ab,t}, \operatorname{do}(X_t=x) \right].

The two values are equal only under a valid identification argument and a correctly estimated identified law.

A policy is

\pi_t \left( x \mid \widehat D_{ab,t}, \mathcal X_t^{\mathrm{cand}} \right).

Let \(\mathfrak P_{\mathrm{adm}}\) be the nonempty admissible policy class, \(\gamma\in[0,1]\), and let \(\mathbb Q_{\Xi}\) and \(\mathbb Q_{\mathcal X}\) be the declared exogenous-path and future candidate-set laws. For a sequence horizon \(H\), the planning objective is

J_{\theta,\mathbb Q_{\Xi},\mathbb Q_{\mathcal X}}(\pi\mid\widehat D_{ab,t}) = \mathbb E_{\theta,\pi,\mathbb Q_{\Xi},\mathbb Q_{\mathcal X}} \left[ \sum_{k=0}^{H-1} \gamma^k u_\tau^{\mathrm{step}} \left( \widetilde D_{ab,t+k+1}, X_{t+k}, \widetilde O_{t+k+1} \right) + \gamma^H V_\tau^{\mathrm{term}} \left( \widetilde D_{ab,t+H} \right) \mid \widehat D_{ab,t} \right].

This is the proper place to assign sequence credit. Do not copy the full eventual transaction reward backward onto every prior message. Use an argmax only when the maximum is attained; otherwise optimize the supremum.

14. Training and off-policy evaluation

For \(L_Y\) primary heads and \(M_Z\) probe heads, minimize

\mathscr J(\theta) = \sum_{\ell=1}^{L_Y} \lambda_\ell^Y \mathscr J_\ell^Y(\theta) + \sum_{m=1}^{M_Z} \lambda_m^Z \mathscr J_m^Z(\theta) + \lambda_{\mathrm{reg}}\Omega(\theta),

with nonnegative weights and \(\Omega(\theta)\ge0\).

Every sign is positive because each term is minimized. Masks, censoring weights, or survival likelihoods belong inside the corresponding head loss.

A slow-vector moving average is valid only when the old and refreshed vectors share one coordinate chart: the encoder is fixed, the new representation is aligned, or the relevant histories are re-encoded after an encoder change.

For one-step off-policy evaluation, let \(\mathcal T_{\mathrm{ope}}\) be the predeclared set of eligible decisions whose utility is mature under the chosen censoring rule, and let \(N_{\mathrm{ope}}=|\mathcal T_{\mathrm{ope}}|>0\). Then

\widehat V_{\mathrm{IPS}}(\pi) = \frac{1}{N_{\mathrm{ope}}} \sum_{t\in\mathcal T_{\mathrm{ope}}} \frac{ \pi_t(x_t\mid\widehat D_{ab,t},\mathcal X_t^{\mathrm{cand}}) }{ \eta_t } u_t^{\mathrm{obs}}.

Here \(u_t^{\mathrm{obs}}\) must be observed and mature, or handled with an appropriate censoring model or weight. If it extends beyond the immediate response, the one-step estimand changes the current proposition under a fixed declared or logged continuation regime. The target-policy numerator must use only pre-decision information. For continuous actions, numerator and denominator are densities under the same dominating measure. Report weight diagnostics and effective sample size. Clipping and self-normalization trade variance for bias or a changed finite-sample estimand; doubly robust methods are often preferable when nuisance models are credible. The estimator requires correct logged probabilities, consistency, overlap, a valid assignment or ignorability argument, correct event ordering, and appropriate treatment of delayed outcomes, censoring, and interference. Learned policies require held-out evaluation or cross-fitting. Sequence policies require sequential estimators; this one-step formula is not stretched across an entire conversation.

15. What the benchmark must establish

The proposed model must be compared against:

prevalence-only prediction;
current-proposition features;
static dyadic tabular features;
shallow history summaries;
a two-tower recommender-style model;
a monolithic sequence model with the same available data.

The evaluation must use future time windows and should separately test:

known people and known companies;
new people inside known companies;
new companies;
new proposition families;
new outcome horizons.

Ablations should remove:

fast state;
slow state;
salesperson state;
relationship state;
composite company aggregation;
source-aware pooling;
salience weighting;
probe heads;
the shared interaction representation.

For a feature family \(f\), under a lower-is-better held-out risk,

\Delta_{\tau,\Delta}^{\mathrm{val}}(f) = \mathcal R_{-f}^{\mathrm{val}} - \mathcal R_{+f}^{\mathrm{val}}.

A positive value means the feature family reduced held-out risk. It does not prove that one learned coordinate is the unique metaphysical feature of the person.

16. Symbols that must not be confused

Symbol	Meaning
\(\nu\)	evolutionary stage
\(\mathbf e_{\varnothing}\)	learned empty-cell representation; not evolutionary stage
\(\tau\)	prediction task
\(H\)	planning length in decision steps
\(\mathcal H\)	history
\(\Pi_{\tau,\Delta}\)	task-summary map
\(\pi_t\)	decision policy
\(\mathbf z_{i,t}\)	fast latent state only
\(\mathbf r_{i,t}^{(\tau)}(x)\)	proposition-conditioned active slice
\(\mathcal A_{\mathrm{corp}}\)	corporate aggregation operator
\(\mathbf Z_t\)	auxiliary probe bundle
\(R_{O,\theta}\)	immediate trace law
\(P_{Y,\theta,\tau,\Delta}\)	delayed outcome law
\(\boldsymbol\Xi\), \(\boldsymbol\xi\)	random exogenous path and a realized/supplied path value
\(\mathbb Q_{\mathcal X}\)	future candidate-set law in planning
\(\mathfrak e\)	declared predictive evaluation regime
\(\mathfrak P_{\mathrm{adm}}\)	admissible policy class
\(\mathscr J(\theta)\)	training objective
\(\mathcal L(X\mid Z)\)	conditional probability law
\(\mathcal P_{s,\theta},\Psi_{s,\theta}\)	individual-state proposition and interaction maps
\(\mathcal P_{D,\theta},\Psi_{D,\theta}\)	dyadic proposition and interaction maps
\(\mathbf d_{i,n}^{\mathrm{rec}}\)	actor-specific record applicability; not memory or a label mask
\(d_{ab,n}^{\mathrm{rel}}\)	relationship-record applicability; zero means the relationship update is the identity
\(a,b\)	salesperson and executive
\(C_a,C_b\)	their corporations

Shortest usable summary

The framework says:

reality contains more possible distinctions than one actor can use;
an actor inherits and develops a way of organizing some of those distinctions;
traces provide evidence about slow organization and fast state;
a proposition is reorganized through the receiving actor rather than merely placed beside them;
the interaction changes an operational state;
immediate and delayed observables are consequences of that transition or trajectory;
propositions can be ranked by forecasted utility;
causal selection requires intervention-grade evidence.

That is the system in its shortest accurate form.